MinT - Best Known (60−37, 60, s)-Nets in Base 27

Best Known (60−37, 60, s)-Nets in Base 27

(60−37, 60, 114)-Net over F₂₇ — Constructive and digital

Digital (23, 60, 114)-net over F₂₇, using

net from sequence [i] based on digital (23, 113)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 23 and N(F) ≥ 114, using
  - an explicitly constructive algebraic function field [i]

(60−37, 60, 163)-Net over F₂₇ — Digital

Digital (23, 60, 163)-net over F₂₇, using

t-expansion [i] based on digital (21, 60, 163)-net over F₂₇, using
- net from sequence [i] based on digital (21, 162)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 21 and N(F) ≥ 163, using
    - a curve from the manYPoints database [i]

(60−37, 60, 172)-Net in Base 27 — Constructive

(23, 60, 172)-net in base 27, using

4 times m-reduction [i] based on (23, 64, 172)-net in base 27, using
- base change [i] based on digital (7, 48, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(60−37, 60, 226)-Net in Base 27

(23, 60, 226)-net in base 27, using

base change [i] based on digital (8, 45, 226)-net over F₈₁, using
- net from sequence [i] based on digital (8, 225)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 8 and N(F) ≥ 226, using
    - a curve from the manYPoints database [i]

(60−37, 60, 14274)-Net in Base 27 — Upper bound on s

There is no (23, 60, 14275)-net in base 27, because

1 times m-reduction [i] would yield (23, 59, 14275)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2 822155 150094 667410 169443 075985 754185 741749 548506 395486 575759 961076 724251 123152 459221 > 27⁵⁹ [i]